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Let Revolution

(x) = x^3 +Ux^2 +Sx + A

, where

U

S

, and

A

are all integers and

U +S + A +1 = 1773

. Given that Revolution has exactly two distinct nonzero integer roots

G

and

B

, find the minimum value of

|GB|

.
Proposed by Jacob Xu
Solution.

\boxed{392}

Notice that

U + S + A + 1

is just Revolution

(1)

so Revolution

(1) = 1773

. Since

G

and

B

are integer roots we write Revolution

(X) = (X-G)^2(X-B)

without loss of generality. So Revolution

(1) = (1-G)^2(1-B) = 1773

1773

can be factored as

32 \cdot 197

, so to minimize

|GB|

we set

1-G = 3

and

1-B = 197

. We get that

G = -2

and

B = -196

|GB| = \boxed{392}

4A